scholarly journals Bhattacharyya Parameter of Monomial Codes for the Binary Erasure Channel: From Pointwise to Average Reliability

Sensors ◽  
2021 ◽  
Vol 21 (9) ◽  
pp. 2976
Author(s):  
Vlad-Florin Drăgoi ◽  
Gabriela Cristescu
Keyword(s):  
Partial Order ◽  
Network Theory ◽  
Error Probability ◽  
Order Relation ◽  
Average Error ◽  
New Order ◽  
Polar Codes ◽  
Order Relations ◽  
Erasure Channel ◽  
Binary Erasure Channel

Monomial codes were recently equipped with partial order relations, a fact that allowed researchers to discover structural properties and efficient algorithm for constructing polar codes. Here, we refine the existing order relations in the particular case of the binary erasure channel. The new order relation takes us closer to the ultimate order relation induced by the pointwise evaluation of the Bhattacharyya parameter of the synthetic channels, which is still a partial order relation. To overcome this issue, we appeal to a related technique from network theory. Reliability network theory was recently used in the context of polar coding and more generally in connection with decreasing monomial codes. In this article, we investigate how the concept of average reliability is applied for polar codes designed for the binary erasure channel. Instead of minimizing the error probability of the synthetic channels, for a particular value of the erasure parameter p, our codes minimize the average error probability of the synthetic channels. By means of basic network theory results, we determine a closed formula for the average reliability of a particular synthetic channel, that recently gain the attention of researchers.

scholarly journals A study on orderings based on fuzzy quasi- order relations

2021 ◽  
Author(s):  
Zhonglin Chai
Keyword(s):  
Partial Order ◽  
Order Relation ◽  
Fuzzy Relation ◽  
Partial Order Relation ◽  
Fuzzy Relations ◽  
Fuzzy Graph ◽  
Feasible Method ◽  
Order Relations ◽  
Finite Set ◽  
Quasi Order

Abstract This paper further studies orderings based on fuzzy quasi-order relations using fuzzy graph. Firstly, a fuzzy relation on a finite set is represented equivalently by a fuzzy graph. Using the graph, some new results on fuzzy relations are derived. In ranking those alternatives, we usually obtain a quasi-order relation, which often has inconsistencies, so it cannot be used for orderings directly. We need to remake it into a reasonable partial order relation for orderings. This paper studies these inconsistencies, and divides them into two types: framework inconsistencies and degree inconsistencies. For the former, a reasonable and feasible method is presented to eliminate them. To eliminate the latter, the concept of complete partial order relation is presented, which is more suitable than partial order relation to rank the alternatives. A method to obtain a reasonable complete partial order relation for a quasi-order relation is given also. An example is given as well to illustrate these discussions. Lastly, the paper discusses the connection between quasi-order relations and preference relations for orderings and some other related problems.

Compact specification of polar codes

2019 ◽  
pp. 40-47
Author(s):  
R. A. Morozov ◽  
P. V. Trifonov
Keyword(s):  
Storage Capacity ◽  
Low Complexity ◽  
Practical Implementation ◽  
Polar Codes ◽  
Practical Relevance ◽  
Memory Consumption ◽  
Full Size ◽  
The Family ◽  
Erasure Probability ◽  
Binary Erasure Channel

Introduction:Practical implementation of a communication system which employs a family of polar codes requires either to store a number of large specifications or to construct the codes by request. The first approach assumes extensive memory consumption, which is inappropriate for many applications, such as those for mobile devices. The second approach can be numerically unstable and hard to implement in low-end hardware. One of the solutions is specifying a family of codes by a sequence of subchannels sorted by reliability. However, this solution makes it impossible to separately optimize each code from the family.Purpose:Developing a method for compact specifications of polar codes and subcodes.Results:A method is proposed for compact specification of polar codes. It can be considered a trade-off between real-time construction and storing full-size specifications in memory. We propose to store compact specifications of polar codes which contain frozen set differences between the original pre-optimized polar codes and the polar codes constructed for a binary erasure channel with some erasure probability. Full-size specification needed for decoding can be restored from a compact one by a low-complexity hardware-friendly procedure. The proposed method can work with either polar codes or polar subcodes, allowing you to reduce the memory consumption by 15–50 times.Practical relevance:The method allows you to use families of individually optimized polar codes in devices with limited storage capacity. 

scholarly journals Homomorphisms and congruences on ωα-bisimple semigroups

1973 ◽  
Vol 15 (4) ◽  
pp. 441-460 ◽  
Author(s):  
J. W. Hogan
Keyword(s):  
Partial Order ◽  
Ordered Set ◽  
Order Relation ◽  
Natural Partial Order ◽  
Partial Order Relation

Let S be a bisimple semigroup, let Es denote the set of idempotents of S, and let ≦ denote the natural partial order relation on Es. Let ≤ * denote the inverse of ≦. The idempotents of S are said to be well-ordered if (Es, ≦ *) is a well-ordered set.

Optimal regular LDPC codes for the binary erasure channel

2005 ◽  
Vol 9 (6) ◽  
pp. 546-548 ◽  
Author(s):  
M. Rashidpour ◽  
A. Shokrollahi ◽  
S.H. Jamali
Keyword(s):  
Ldpc Codes ◽  
Erasure Channel ◽  
Binary Erasure Channel
Sign in / Sign up

Export Citation Format

Share Document